Factor by splitting the middle term 12x^2+17x+6?

Answer 1

12x^2+17x+6 = (12x^2+9x)+(8x+6)
=3x(4x+3)+2(4x+3)
=(3x+2)(4x+3) - PK

Answer 2

To factor this you need factors of 12 and factors of six that when multiplied and added = 17. The factors of 12 are 12, 6, 4, 3, 2, 1 and the factors of 6 are 6, 3, 2, 1. By trial and error you can get 12x^2 + 17x + 6 = (4x + 3)(3x + 2).

Proof: (4x + 3)(3x + 2) ==> 4x(3x + 2) + 3(3x + 2)
==> 4x(3x) + 4x(2) + 3(3x) + 3(2) ==> 12x^2 + 8x + 9x + 6
==> 12x^2 + 17x + 6

Answer 3

The trick here is to try to find a factor which is common to more than one grouping of terms.

If we split up the above expression as follows, and do a little thinking and paper scratch work, we see that there is a common factor:

12x^2+8x+9x+6 = (12x^2+8x)+(9x+6).

Now factor each grouping of terms separately.

(12x^2+8x) = 4x(3x+2)
(9x+6) = 3(3x+2)

Now all we have to do is put the parts back together again and extract the appropriate common factor(s).

12x^2+8x+9x+6 = 4x(3x+2)+ 3(3x+2) = (4x+3)(3x+2).

Notice that if we actually multiplied the last two factors out longhand, we will get the initial expression.

Answer 4

- short answer :
12x^2+17x+6=12x^2+(8x+9x)+6=(12x^2+8x)+(9x+6)=4x(3x+2)+3(3x+2)=(3x+2)(4x+3)=12(x+2/3)(x+3/4)
- long answer if you wanna know more :
The example was built to be an easy one to be solved by splitting the middle term.
This is a degree 2 equation, which generally looks like this : f(x)=ax^2+bx+c. in your case a=12,

b=17 and c=6. However, the most common thing to do with an equation is to solve it, so we may say

: for what "x" f(x)=0 ? If f(x)=0, also g(x)f(x)=0, whoever g(x) is....
Solving the degree 2 equation by splitting the middle term is the easiest way to do it, but it is

not always possible to split.... here, split this : x^2+1.416666x+0.5
Believe me, this is "THE SAME EXAMPLE", so what is beyond splitting ??
First you must know how to solve a degree 2 equation that generally is f(x)=ax^2+bx+c; you follow

this steps:
1. calculate delta=b^2-4ac; for you : delta=17^2-4x12x6=289-288=1
2. calculate sqrt(delta) if delta>=0. If delta is negative you are dealing with complex numbers,

but that's another story......so sqrt(delta)=sqrt(1)=1, and -1. This is something that few people

know: there are 2 square roots for one given number, and this roots are equal in absolute value,

but they differ by sign. Also sqrt(9)=3 and -3, because (-3)^2=9 get it?.......
3. if delta>=0, than you calculate the values "x" for which f(x)=ax^2+bx+c=0, just like this :

x=(-b+sqrt(delta))/(2a), which will generate 2 values(REMEMBER!! because sqrt(delta) generates 2

values), so x=(-17+sqrt(1))/(2x12); first value is (-17-1)/24=-18/24=-3/4; second value is

(-17+1)/24=-16/24=-2/3. This 2 values are x1 and x2, also known as the roots of f(x).
4. if you have the roots, you may wright f(x) like this : f(x)=a(x-x1)(x-x2). so

12(x-(-2/3))(x-(-3/4))=12(x+2/3)(x+3/4)
Now remember when i sad if f(x)=0, also g(x)f(x)=0 ?? please observe that if x=x1 or x=x2, your

equation is 0(zero), because one or another of those parenthesis (x+2/3)or(x+3/4) goes to 0. So

that "12" doesn't matter. So you may put whtvr number you wanna in the place of 12. The example i

asked you to "split" : x^2+1.416666x+0.5 is actually your example without the "12".
Now if you remember, i sad "The example was built to be an easy one to be solved by splitting the

middle term" : if you sum your roots by fractional laws: x1+x2=2/3+3/4 you must multiply the 1st

one with 4, and the 2nd with 3, and you will have : 8/12+9/12=17/12. Do you see now: 17 is the

term to split....and the entire equation was multiply by "12" so you don't have to split a

fraction.So this is how you built "easy to split" examples.
Going even Dipper, if x1 and x2 are the roots of ax^2+bx+c, x1+x2=-b/a and x1*x2=c/a, so "b" is

kind of their sum, and "c" is kind of their prod....These are the "Viette" relations....

Answer 5

12x^2+17x+6=12x^2+8x+9x+6
=4x(3x+2)+3(3x+2)
=(3x+2)(4x+3) Ans

Answer 6

12x^2+9x+8x+6
3x(4x+3)+2(4x+3)
=(4x+3)(3x+2)

Answer 7

(3x + 2)(4x + 3)